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Title: Polynomial ideals, association schemes, and the Q-polynomial property

Abstract:

Let X ⊆ S^{m−1} be a spherical code in C^m. We study the ideal I ⊆ C[z_1, . . . , z_m] of polynomials that vanish on the points of X: I = { F(z) | (∀a ∈ X) (F(a) = 0) }. The primary example of interest is where the Gram matrix of X is proportional to the first idempotent in some Q-polynomial ordering of an association scheme (X, R) defined on X. We will discuss examples ranging from the Leech lattice to posets to Paley graphs. I will present two “dual girth” parameters γ_1(X) and γ_2(X); I conjecture that, except when (X, R) is the association scheme of a polygon, γ_2(X) ≤ 6. I see this as dual to a conjecture about the fundamental group of a distance-regular graph.